Notations T

$T_{\NVar{n}}$: tangent numbers; §24.15(ii)
$T\left(\NVar{z}\right)$: Tree $T$ -function; §4.13
$T^{*}_{\NVar{n}}\left(\NVar{x}\right)$: shifted Chebyshev polynomial of the first kind; Table 18.3.1
$\tan\NVar{z}$: tangent function; (4.14.4)
$\tanh\NVar{z}$: hyperbolic tangent function; (4.28.4)
$\tau\left(\NVar{n}\right)$: Ramanujan’s tau function; (27.14.18)
$\vartheta\left(\NVar{x}\right)=\theta_{3}\left(0,x\right)$: alternative notation; (27.13.4)
(with $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function)
$\theta\left(\NVar{z}\right)$: Airy phase function; (9.8.4)
$\theta_{\NVar{\nu}}\left(\NVar{x}\right)$: phase of Bessel functions; (10.18.3)
$\Theta(\NVar{\boldsymbol{{\phi}}}|\NVar{\mathbf{B}})=\theta\left(\boldsymbol{{% \phi}}/(2\pi i)\middle|\mathbf{B}/(2\pi i)\right)$: notation used by Belokolos et al. (1994), Dubrovin (1981); §21.1
(with $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function, $\pi$ : the ratio of the circumference of a circle to its diameter and $\mathrm{i}$ : imaginary unit)
$\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$: Riemann theta function; (21.2.1)
$\Theta(\NVar{z}|\NVar{\tau})=\theta_{4}\left(u\middle|\tau\right)$: Jacobi’s notation; §20.1
(with $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function)

$\Theta_{1}(\NVar{z}|\NVar{\tau})=\theta_{3}\left(u\middle|\tau\right)$: Jacobi’s notation; §20.1
(with $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function)
$\theta_{c}(\NVar{z}|\NVar{\tau})=\ifrac{\theta_{2}\left(u\middle|\tau\right)}{% \theta_{2}\left(0\middle|\tau\right)}$: notation used by Neville (1951); §20.1
(with $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function)
$\theta_{d}(\NVar{z}|\NVar{\tau})=\ifrac{\theta_{3}\left(u\middle|\tau\right)}{% \theta_{3}\left(0\middle|\tau\right)}$: notation used by Neville (1951); §20.1
(with $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function)
$\hat{\theta}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$: scaled Riemann theta function; (21.2.2)
$\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function; §20.2(i)
$\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function; §20.2(i)
$\vartheta_{\NVar{j}}(\NVar{z}|\NVar{\tau})=\theta_{j}\left(\pi z\middle|\tau\right)$: notation used by McKean and Moll (1999, p. 125); §20.1
(with $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function and $\pi$ : the ratio of the circumference of a circle to its diameter)
${\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$: phase of Coulomb functions; (33.2.9)
$\theta_{n}(\NVar{z}|\NVar{\tau})=\ifrac{\theta_{4}\left(u\middle|\tau\right)}{% \theta_{4}\left(0\middle|\tau\right)}$: notation used by Neville (1951); §20.1
(with $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function)
$\theta_{s}(\NVar{z}|\NVar{\tau})=\pi{\theta_{3}}^{2}\left(0\middle|\tau\right)% \ifrac{\theta_{1}\left(u\middle|\tau\right)}{\theta_{1}'\left(0\middle|\tau% \right)}$: notation used by Neville (1951); §20.1
(with $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function and $\pi$ : the ratio of the circumference of a circle to its diameter)
$\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$: Riemann theta function with characteristics; (21.2.5)
$\operatorname{tr}\NVar{\mathbf{A}}$: trace of matrix; (1.2.62)

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